How can I optimize coefficients of a polynomial using fmincon?

I have 3 equations that represents a curve in 3D space. These equations are

X(t) = at^5 + bt^4 + ct^3 + dt^2 + et + f 
Y(t) = at^5 + bt^4 + ct^3 + dt^2 + et + f 
Z(t) = at^5 + bt^4 + ct^3 + dt^2 + et + f

I have certain constraints for this problem. But figuring out how to write this into a function in Matlab is difficult. I don’t know what to do with the t^n. Since the parameters that are changing are the coefficients, how do I write this into Matlab? My cost function is J= sqrt( dxdt^2 + dydt^2 + dzdt^2) (Or at least, this is what I have been interpreting it as. The overall goal is to minimize the length of the cost function )

[Matt J]

These equations are...

As you've written it, the same coefficients are used for X,Y, and Z meaning your curve has to be a straight line.

My cost function is J= sqrt( dxdt^2 + dydt^2 + dzdt^2)

This is not a scalar-valued cost function. Did you really mean,

J= sum( dxdt^2 + dydt^2 + dzdt^2)^2 

[Bryant Springle]

Sorry, I should have mentioned that teh coefficients are different for each equation. I just wanted to show that each equation is modeled by a quintic polynomial. And after viewing another answer, I am now realizing that the cost function is as integral

Answer 1

[Bruno Luong]

"The overall goal is to minimize the length of the cost function "

The overall length is

L = integral sqrt(dx/dt^2+dy/dt^2+dy/dt^2)(t) dt

t is then a integration variable to compute L that you want ti minimize.

You should program the code to compute L from coefficients by integrating on t, over whatever the interval that defines your curve.

[Bryant Springle]

Bruno.... I dont know why I didnt recognize that my cost function was an integral. That makes a lot of sense. I do have another question. My equality constraints and inequality constraints state that:

From the first image, it says that at those specific times, those are the x,y, and z coordinates. How would you suggest I do this in the constraint function for fmincon? Heres how I have been thinking of writing it. (I dont think this would work, so suggestions are very welcomed)

Here is an example for how I would write the initial waypoint constriant for x at t = 0.

x(t) = at^5 + bt^4 + ct^3 + dt^2 + et + f

If I put all of my coeficients into one long vector, X, I get a 18 x 1 vector, where the first 6 are the coefficients for x.

At time t = 0, f should equal 5. So I was thinking the equality constraint should be

[X(6) - 5];

Would this be a valid constraint?

[Bruno Luong]

"Would this be a valid constraint?"

Yes.

You can use Aeq/beq arguments as well since value constraints us a linear constraint wrt coefficients.